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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-10 09:22:27 UTC</tt>.<br>
: The original revision id was <tt>397272172</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]


The hemimean clan, tempering out the no-threes hemimean comma 3136/3125, contains the temperaments misty and semisept, considered below, as well as meantone, hemiwuerschmidt, parakleismic, hemithirds, clyde, hemischismic, passion, hexe and sycamore considered elsewhere.
The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.  


Comma: 3136/3125
Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.


No-threes [[POTE tuning|POTE generator]] (~28/25): 193.772
Didacus has canonical subgroup extensions to primes 11 and 13, at [[#Undecimal didacus|undecimal didacus]]. Other subgroup extensions include rectified hebrew and isra.


Map: [&lt;1 0 0 -3|, &lt;0 0 2 5|]
Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. Discussed elsewhere are
EDOs: [[6edo|6]], [[12edo|12]], [[19edo|12]], [[31edo|31]], [[68edo|68]], [[80edo|80]], [[87edo|87]], [[99edo|99]], [[130edo|130]], [[217edo|217]], [[229edo|229]], [[328edo|328]], [[347edo|347]], [[446edo|446]], [[545edo|545c]], [[675edo|675]]
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Mavila family #Mohavila|Mavila family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Bidia]]'' (+2048/2025) → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hexe]]'' (+50/49 or 128/125) → [[Augmented family #Hexe|Augmented family]]
* [[Misty]] (+5120/5103) → [[Misty family #Septimal misty|Misty family]]
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* [[Parakleismic]] (+4375/4374) → [[Parakleismic family #Parakleismic|Parakleismic family]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Decistearn]]'' (+118098/117649) → [[Trisedodge family #Decistearn|Trisedodge family]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]


=Misty=
= 2.5.7 subgroup =
Commas: 3136/3125, 5120/5103
== Didacus ==
{{main|Didacus}}


[[POTE tuning|POTE generator]]: 703.021
See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].


Map: [&lt;3 0 26 56|, &lt;0 1 -4 -10|]
[[Subgroup]]: 2.5.7
Wedgie: &lt;&lt;3 -12 -30 -26 -56 -36||
EDOs: [[12edo|12]], [[87edo|87]], [[99edo|99]]


==11-limit==
[[Comma list]]: [[3136/3125]]
Commas: 441/440, 3136/3125, 4375/4356


[[POTE tuning|POTE generator]]: 703.210
{{Mapping|legend=3| 1 0 -3 | 0 2 5 }}


Map: [&lt;3 0 26 56 77|, &lt;0 1 -4 -10 -14|]
: sval mapping generators: ~2, ~56/25
EDOs: 12, 87, 99e, 186e


==13-limit==
{{Mapping|legend=5| 1 0 0 -3 | 0 0 2 5 }}
Commas: 196/195, 352/351, 364/363, 625/624


[[POTE tuning|POTE generator]]: 703.303
: [[gencom]]: [2 56/25; 3136/3125]


Map: [&lt;3 0 26 56 77 92|, &lt;0 1 -4 -10 -14 -17|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772
EDOs: 12, 87, 186ef, 273ef


=Mystic=
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
Commas: 325/324, 441/440, 896/891, 1001/1000


[[POTE tuning|POTE generator]]: 703.311
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents


Map: [&lt;3 0 26 56 77 -46|, &lt;0 1 -4 -10 -14 12|]
[[Badness]] (Sintel): 0.091
EDOs: 12, 87, 186e, 273e, 633bde


==17-limit==
= Strong extensions =
Commas: 196/195, 352/351, 364/363, 625/624
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong extensions
|-
! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*
|-
! Temperament !! 5-limit generator location
|-
| [[#Hemiwürschmidt|Hemiwürschmidt]] || [[Würschmidt family#Würschmidt|Würschmidt]] || +2 || +16 || ↓ [[31edo|31]]
|-
| [[#Hemithirds|Hemithirds]] || [[Luna family#Luna|Luna]] || +1 || -15 || ↑ 31 <br /> ↓ [[25edo|25]]
|-
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
<nowiki />* Defined by intersection with other documented extensions


[[POTE tuning|POTE generator]]: 703.233
== Hemiwürschmidt ==
''[[#Strong extensions|Return to the map]]''


Map: [&lt;3 0 26 56 77 92 -2|, &lt;0 1 -4 -10 -14 -17 3|]
{{See also| Würschmidt family }}
EDOs: 87, 186ef, 273ef


==19-limit==
'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.
Commas: 154/153, 196/195, 245/243, 273/272, 286/285, 325/324


[[POTE tuning|POTE generator]]: 703.253
[[Subgroup]]: 2.3.5.7


Map: [&lt;3 0 26 56 77 92 -2 8|, &lt;0 1 -4 -10 -14 -17 3 1|]
[[Comma list]]: 2401/2400, 3136/3125
EDOs: 46, 87, 133, 326de, 413degh


=Semisept=
{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}
Comma: 782757789696/762939453125


POTE generator: ~192/125 = 735.146
Mapping generators: ~2, ~25/14


Map: [&lt;1 12 6|, &lt;0 -17 -6|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
EDOs: 18, 31, 80, 111
Badness: 0.6306


==7-limit==
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
Commas: 3136/3126, 1728/1715


[[POTE tuning|POTE generator]]: (~75/49) 735.155
[[Badness]]: 0.020307


Map: [&lt;1 12 6 12|, &lt;0 -17 -6 -15|]
=== 2.3.5.7.23 subgroup ===
Wedgie: &lt;&lt;17 6 15 -30 -24 18||
As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).
EDOs: [[18edo|18]], 31, [[80edo|80]], 111


==11-limit==
Subgroup: 2.3.5.7.23
Commas: 176/175, 540/539, 1331/1323


[[POTE tuning|POTE generator]]: (~55/36) 735.125
[[Comma list]]: 576/575, 736/735, 1127/1125


Map: [&lt;1 12 6 12 20|, &lt;0 -17 -6 -15 -27|]
{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}
EDOs: 18, 31, 80, 111, 364cd,
Badness: 0.0225


===Semishly===
Mapping generators: ~2, ~25/14
Commas: 144/143, 176/175, 196/195, 275/273


POTE generator: ~13/10 = 464.980
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901


Map: [&lt;1 12 6 12 20 8|, &lt;0 -17 -6 -15 -27 -7|]
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
EDOs: 31, 49f, 80f
Badness: 0.0284


==13-limit==
Badness (Sintel): 0.304
Commas: 176/175, 351/350, 540/539, 1375/1372


[[POTE tuning|POTE generator]]: 735.126
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 12 6 12 20 -11|, &lt;0 -17 -6 -15 -27 24|]
Comma list: 243/242, 441/440, 3136/3125
EDOs: 31, 80, 111
Badness: 0.0252


==17-limit==
{{Mapping|legend=0| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
Commas: 176/175, 256/255, 351/350, 640/637, 715/714


[[POTE tuning|POTE generator]]: (~26/17) 735.125
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840


Map: [&lt;1 12 6 12 20 -11 -10|, &lt;0 -17 -6 -15 -27 24 23|]
{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}
EDOs: 31, 80, 111


==19-limit==
Badness: 0.021069
Commas: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


[[POTE tuning|POTE generator]]: 735.116
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 12 6 12 20 -11 -10 -8|, &lt;0 -17 -6 -15 -27 24 23 20|]
Comma list: 243/242, 351/350, 441/440, 3584/3575
EDOs: 31, 80, 111


==23-limit==
{{Mapping|legend=0| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
Commas: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


[[POTE tuning|POTE generator]]: 435.106
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829


Map: [&lt;1 12 6 12 20 -11 -10 -8 18|,  
{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
&lt;0 -17 -6 -15 -27 24 23 20 -22|]
EDOs: 31, 80, 111, 191cdh, 302cdgh


=Sengagen=
Badness: 0.023074
Commas: 3136/3125, 420175/419904


POTE generator: ~686/675 = 24.217
==== Hemithir ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 2 2|, &lt;0 29 16 40|]
Comma list: 121/120, 176/175, 196/195, 275/273
Wedgie: &lt;&lt;29 16 40 -42 -18 48||
EDOs: 49, 50, 99, 248, 347, 446
Badness: 0.0580


=Subpental=
{{Mapping|legend=0| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
Commas: 3136/3125, 19683/19600


POTE generator: ~56/45 = 378.467
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918


Map: [&lt;1 6 8 17|, &lt;0 -14 -18 -45|]
{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
Wedgie: &lt;&lt;14 18 45 -4 32 54||
EDOs: 19, 111, 130, 929c, 1059c, 1189bc, 1319bc
Badness: 0.0543


==11-limit==
Badness: 0.031199
Commas: 540/539, 3136/3125, 8019/8000


POTE generator: ~56/45 = 378.440
=== Hemiwur ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 6 8 17 -6|, &lt;0 -14 -18 -45 30|]
Comma list: 121/120, 176/175, 1375/1372
EDOs: 19, 111, 130, 241, 371ce, 501cde, 872cde
Badness: 0.0453


==13-limit==
{{Mapping|legend=0| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
Commas: 351/350, 540/539, 676/675, 3136/3125


POTE generator: ~56/45 = 378.437
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884


Map: [&lt;1 6 8 17 -6 16|, &lt;0 -14 -18 -45 30 -39|]
{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
EDOs: 19, 111, 130, 241, 371ce
Badness: 0.0239


=Arch=
Badness: 0.029270
Commas: 3136/3125, 5250987/5242880


POTE generator: ~64/63 = 27.668
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 2|, &lt;0 -18 14 35|]
Comma list: 121/120, 176/175, 196/195, 275/273
Wedgie: &lt;&lt;18 -14 -35 -64 -106 -42||
EDOs: 43, 87, 130, 217, 347, 824c, 1171c, 1518cd
Badness: 0.0943


==11-limit==
{{Mapping|legend=0| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
Commas: 441/440, 3136/3125, 4000/3993


POTE generator: ~64/63 = 27.663
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004


Map: [&lt;1 2 2 2 3|, &lt;0 -18 14 35 20|]
{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
EDOs: 43, 87, 130, 217, 347e, 911cde
Badness: 0.0365


==13-limit==
Badness: 0.028432
Commas: 364/363, 441/440, 676/675, 3136/3125


POTE generator: ~64/63 = 27.660
==== Hemiwar ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 2 2 3 4|, &lt;0 -18 14 35 20 -13|]
Comma list: 66/65, 105/104, 121/120, 1375/1372
EDOs: 43, 87, 130, 217, 347e, 564e
Badness: 0.0195


=Emka=
{{Mapping|legend=0| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
Commas: 3136/3125, 84035/82944


POTE generator: ~48/35 = 551.782
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698


Map: [&lt;1 14 6 12|, &lt;0 -27 -8 -20|]
{{Optimal ET sequence|legend=1| 6f, 31 }}
EDOs: 37, 50, 87, 137d, 224d
Badness: 0.1443


==11-limit==
Badness: 0.044886
Commas: 385/384, 2401/2376, 3136/3125


POTE generator: ~11/8 = 551.765
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  


Map: [&lt;1 14 6 12 3|, &lt;0 -27 -8 -20 1|]
The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].
EDOs: 37, 50, 87, 224d, 311d
Badness: 0.0547


==13-limit==
Subgroup: 2.3.5.7.11
Commas: 196/195, 364/363, 385/384, 625/624


POTE generator: ~11/8 = 551.758
Comma list: 2401/2400, 3025/3024, 3136/3125


Map: [&lt;1 14 6 12 3 6|, &lt;0 -27 -8 -20 1 -5|]
{{Mapping|legend=0| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
EDOs: 37, 50, 87, 224d, 311d, 398d
Badness: 0.0297


: mapping generators: ~2, ~147/110


</pre></div>
Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Hemimean clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:48:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;
{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
 
Badness: 0.034814
 
=== Semihemiwür ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3136/3125, 9801/9800
 
{{Mapping|legend=0| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
 
: mapping generators: ~99/70, ~495/392
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
 
Badness: 0.044848
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
 
{{Mapping|legend=0| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
 
Badness: 0.023388
 
===== Semihemiwürat =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
 
{{Mapping|legend=0| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
 
Badness: 0.028987
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
 
{{Mapping|legend=0| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
 
{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
 
Badness: 0.021707
 
===== Semihemiwüram =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
 
{{Mapping|legend=0| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
 
{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
 
Badness: 0.029718
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
 
{{Mapping|legend=0| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428
 
{{Optimal ET sequence|legend=1| 62egh, 68
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